Six variations on fair wages and the long-run Phillips curve

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Working Paper Series Department of Economics

University of Verona

Six variations on fair wages and the long-run Phillips curve

Andrea Vaona

WP Number: 17 November 2010

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Six variations on fair wages and the Phillips curve

Andrea Vaona

University of Verona (Department of Economic Sciences), Via dell’Artigliere 19, 37129 Verona, Italy. E-mail: andrea.vaona@univr.it.

Phone: +390458028537

Kiel Institute for the World Economy

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Six variations on fair wages and the Phillips curve

Abstract

The present paper explores the connection between in‡ation and unemployment in di¤erent models with fair wages both in the short and in the long runs. Under customary assumptions regarding the sign of the parameters of the e¤ort function, more in‡ation lowers the unemployment rate, though to a declining extent. This is because …rms respond to in‡ation - that spurs e¤ort by decreasing the reference wage - by increasing employment, so to maintain the e¤ort level constant, as implied by the Solow condition. Under wage staggering this e¤ect is stronger because wage dispersion magni…es the impact of in‡ation on e¤ort. A stronger e¤ect of in‡ation on unemployment is also produced under varying as opposed to …xed capital, given that in the former case the boom produced by a monetary expansion is reinforced by an increase in investment. Our baseline results are robust to the adoption of a model based on reciprocity in labour relations. Therefore, we provide a new theoretical foundation for recent empirical contributions …nding negative long- and short-run e¤ects of in‡ation on unemployment.

Keywords: e¢ ciency wages, money growth, long-run Phillips curve, trend in‡ation, wage staggering, reciprocity in labour relations.

JEL classi…cation codes: E3, E20, E40, E50.

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1 Introduction

The economic literature has recently witnessed a ‡ourishing of contributions nesting an e¢ ciency wages framework into business cycle models. Earlier models were proposed within the real business cycle (RBC) realm. Danthine and Donaldson (1990), for instance, showed that e¢ ciency wages within a RBC model can produce structural unemployment, but not wage stickiness over the economic cycle. With di¤erence to Danthine and Donaldson (1990), which focused on a gift exchange model, Uhlig and Xu (1995) and Gomme (1999) adopted a shirking model. However, in a rather similar way, they found that wages tend to be too volatile and employment not enough so over the cycle. In Kiley (1997) e¢ ciency wages generate completely a-cyclical real wages, but not a greater endogenous price stickiness, because the a-cyclical real-wage requires countercyclical e¤ort and hence a procyclical marginal cost.

Collard and de la Croix (2000) showed that, once including past compen- sations into the reference wage, an e¢ ciency wages/RBC model can replicate wage acyclicality. Along similar lines, Danthine and Kurmann (2004) proposed a model combining e¢ ciency wages of the gift exchange variety - also termed fair wages - with sticky prices, showing that it can well account for the low correlation between wages and employment, also displaying a greater internal propagation of monetary shocks than standard New Keynesian models. Danthine and Kurmann (2008), inspired by Rabin (1993), explicitly mod- elled the psychological bene…ts arising from gift exchanges between …rms and workers in terms of remuneration and e¤ort respectively. Danthine and Kur- mann (2010) incorporated a reciprocity-based model of wage determination into a dynamic general equilibrium model, which was then estimated on U.S.

data. They highlighted that wage setting is driven more by rent-sharing and past wages, than by aggregate employment conditions.

Alexopoulos (2004, 2006, 2007) developed a model in which shirkers are not dismissed once detected. They, instead, forgo an increase in compensa-

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tion. Under these assumptions it was showed that an e¢ ciency wage model can well replicate empirical evidence regarding the response of the economic system to technological, …scal and monetary shocks.

The present paper, instead, focuses on the long-run and short-run implications of e¢ ciency wages for the connection between unemployment and in‡ation under trend money growth within a dynamic general equilibrium framework. In so doing, we extend a literature that so far investigated the long-run and, to a lesser extent, the short-run e¤ects of money growth by resorting only to models with wage/price stickiness. Pioneering contributions on this issue were King and Wolman (1996) and Ascari (1998). The former study considered a model with a shopping time technology and it obtained a number of di¤erent results, among which there is that long-run in‡ation reduces …rms’markup, boosting the level of output. Ascari (1998), instead, showed that in wage-staggering models money can have considerable negative non-superneutralities once not considering restrictively simple utility and production functions. Deveraux and Yetman (2002) focused on a menu cost model. An analysis of dynamic general equilibrium models under di¤erent contract schemes in presence of trend in‡ation was o¤ered in Ascari (2004).

Graham and Snower (2004), instead, examined the microeconomic mechanisms underlying this class of models. In presence of Taylor wage staggering, in a monopolistically competitive labour market, they highlighted three channels through which in‡ation a¤ects output: employment cycling, labour supply smoothing and time discounting. The …rst one consists in …rms con- tinuously shifting labour demand from one cohort to the other according to their real wage. Given that di¤erent labour kinds are imperfect substitutes, this generates ine¢ ciencies and it tends to create a negative in‡ation-output nexus. The second one is that households demand a higher wage in presence of employment cycling given that they would prefer a smoother working time. This decreases labor supply and aggregate output. Finally under time discounting the contract wage depends more on the current (lower) level of

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prices than on the future (higher) level of prices and, therefore - over the contract period - the real wage will be lower the greater is the in‡ation rate, spurring labour demand and aggregate output. The time discounting e¤ect dominates at lower in‡ation rates, while the other two e¤ects at higher in‡a- tion rates, producing a hump-shaped long-run Phillips curve. The ultimate goal of Graham and Snower (2004) is questioning the customary assumption to identify aggregate demand and supply shocks, namely that the former ones would be temporary and the latter ones not so. As a consequence also the concept of the NAIRU would be unsuitable for a fruitful investigation of the dynamics of the unemployment rate.

Graham and Snower (2004) was extended in a number of di¤erent di- rections. Graham and Snower (2008) showed that under hyperbolic time discounting positive money non-superneutralities are more sizeable than under exponential discounting. Vaona and Snower (2007, 2008) showed how the shape of the long-run Phillips curve depends on the shape of the production function. Finally, Vaona (2010) extended the model by Graham and Snower (2004) from the in‡ation-output domain to the in‡ation-real growth one.

We here propose six variations on the theme of e¢ ciency wages and the Phillips curve. In the …rst one, e¢ ciency wages of the gift exchange variety are coupled with trend money growth, once specifying the reference wage as a function of the unemployment rate, the current individual real wage, the current aggregate real wage and of the current real value of the past aggregate wage. After Becker (1996), this speci…cation has been termed in the literature as social norm case. Being here the reference wage a function of the current real value of the past aggregate wage and not, as in Danthine and Kurmann (2004), of the past real wage, we can highlight the macroeconomic consequences of a peculiar gift exchange between …rms and workers that was not investigated so far, though being empirically relevant. Bewley (1999) stresses many times that …rms, though not liking wage indexation, are not insensitive to the damages produced by in‡ation to the purchasing power of

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wages. If workers perform well, pay managers will consider fair to o¤set the negative e¤ect in‡ation can have on workers’standard of living¹. This can be conceptualized as a gift exchange: workers elicit e¤ort and …rms maintain the purchasing power of their wages. We show that this mechanism can produce sizeable money non-superneutralities both in the short and long run.

In our second variation, the reference wage is not a function of the current real value of the past aggregate wage, rather of that of the past individual one, as in the personal norm case. Our third model combines Taylor wage stickiness with fair wages of the social norm variety. In this setting, positive money non-superneutralities turn out to be stronger than under ‡exible wages². The fourth variation extends the …rst one by considering varying instead of …xed capital. In the …fth and sixth variations, we show that our baseline results also hold in a framework à la Danthine and Kurmann (2008, 2010).

With di¤erence to Graham and Snower (2004, 2008) we provide not only a long-run analysis but also a short-run one, because we think that, even if

1To the reader convenience we report some quotations from Bewley (1999). "Other important in‡uences were raises at other …rms competing in the same labor markets and changes in the cost of living. Employers wished to protect employees’standard of living, both to maintain morale and out of a sense of moral responsibility. Many …rms did not, however, fully o¤set increases in living costs in all circumstances" (pp. 160-161). "When hiring someone, I pay them a salary equal to the value of their job. In‡ation e¤ectively reduces it, and fairness requires that I o¤set the reduction. I think that is the way it ought to be. If I hire people at a certain rate, I want to keep that level constant in terms of standard of living" (p. 164). "In deciding on the level of raises, we look at the rate of in‡ation in the cost of living. It is an indicator of what the competition is doing (...)" (p. 165). "Cost-of-living in‡ation was a major factor in the determination of raises.

[...] The pay of low-performing workers was often allowed to fall behind in‡ation" (p.

208). "Question: Would a pay cut of 10 percent with no in‡ation have more impact on employees than a pay freeze with 10 percent in‡ation? Answer: Both are wage cuts. [...]

The company would have to be in trouble. In both cases, people might leave [...]" (p.

209).

Also Levine (1993) …nds that companies tend to o¤er larger wage increases in presence of higher in‡ation, though not in a one-to-one proportion.

2Fan (2007) proposed to merge sticky and e¢ ciency wages, but not in an intertemporal optimization framework as we do here.

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one cannot identify demand and supply shocks on the basis of their tran- sience, it will be interesting to investigate how the economic system reacts to temporary monetary shocks. In other words, transition dynamics does not lose interest.

Our results can o¤er a new theoretical foundation for the empirical …nd- ings obtained in various recent contributions, that have already been discussed in Karanassou et al. (2010). A brief review is o¤ered here focusing on the analyzed countries and time periods, on the adopted econometric methods and on a common result of theirs, which is particularly relevant to our analysis.

Karanassou et al. (2003, 2005) bring dynamic multi-equation models to both European and US annual data from 1977 to 1998 and from 1966 to 2000 respectively. In the former case they rely on panel data methods, while in the latter one on the three-stage least squares (3SLS) estimator. Karanassou et al. (2008a) expands the model by Karanassou et al. (2005) by endo- genizing productivity and …nancial wealth and deriving the unemployment rate from labour supply and demand equations. Then they apply a six- equation structural model to US data running from 1965 to 2000 by using an autoregressive distributed lags (ARDL) estimator. Model simulation are

…nally o¤ered over the period from 1993 to 2000 reaching the conclusion that money growth put upward pressure on in‡ation and substantially lowered unemployment. Rising productivity growth, budget de…cit reductions, and a widening trade de…cit played a minor role in in‡ation and unemployment dynamics. Karanassou et al. (2008b) bring a structural model to Spanish annual data from 1966 to 1998 by using both ARDL and 3SLS estimators.

A common result of theirs is that in‡ation and unemployment are connected not only in the short-run but in the long-run too. The long-run elasticity of in‡ation with respect to unemployment was estimated to be about 3:5;

which was explained by resorting to frictional growth, namely the interplay between frictions (lagged adjustments) and growth in economic variables. In

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the light of our models, this result can be also interpreted as the outcome of e¢ ciency wages mechanisms as explained below.

The rest of this paper is structured as follows. The next section introduces the households’ problem and the government budget constraint, which are common to most of the models here presented. Afterwards, we will introduce the …rms’problem for the social norm case with ‡exible wages, the personal norm case under ‡exible wages, the social norm case with wage staggering and the social norm case with varying capital. The seventh section shows that our results hold also adopting a model based on reciprocity on labour relations à la Danthine and Kurmann (2008, 2010). The last section concludes.

In all the cases, we show what is the impact of money growth on both the unemployment and the in‡ation rates both in the short- and in the long-runs and we discuss the plausibility of our models in order to detect our preferred ones. Introducing capital accumulation at a later stage is not an unusual procedure in the New-Keynesian literature (see for instance Huang and Liu, 2002; Ascari, 2004; Danthine and Kurmann, 2010). Some contributions do not even consider capital accumulation (Ascari 1998; Graham and Snower, 2004, 2008; Danthine and Kurmann, 2008; Ascari and Ropele, 2009). This can be explained by at least two reasons. In the …rst place, as reminded by Ascari (2004), McCallum and Nelson (1999) argued that it is di¢ cult to specify a capital demand function which is "both analytically tractable and empirically successful". In the second place - similarly to sticky wages/prices models (Ascari, 2004, Vaona, 2010) - the core of our model is in the labour market and capital accumulation turns out to be just a superstructure, not inducing any qualitative change in our results. Therefore, we believe our exposition strategy is the most suited to convey the underlying intuition of our model.

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2 The households’ problem and the govern- ment budget constraint

We follow Danthine and Kurmann (2004, 2008, 2010), by supposing the economy to be populated by a continuum of households normalized to 1, each composed by a continuum of individuals also normalized to 1. We adopt a money-in-the-utility-function approach to preserve comparability with the trend in‡ation literature (Ascari 2004, Graham and Snower, 2004, 2008)³. Households maximize their discounted utility

max

fct+i(h);Bt+i(h);Mt+i(h);et+i(h)g

X1 i=0

t+iE U

( c_t+i(h) ; n_t+i(h) G [e_t+i(h)] ; V h

Mt+i(h) Pt+i

i

)!

(1) subject to a series of income constraints

c_t+i(h) = W_t+i(h)

P_t+i n_t+i(h) + T_t+i(h) P_t+i

M_t+i(h) P_t+i + +M_{t+i 1}(h)

P_t+i

B_t+i(h)

P_t+i +B_{t+i 1}(h)

P_t+i ^t+i+ q_t+i(h) (2) where is the discount factor, E is the expectation operator, U is the utility function, ct+i(h)is consumption of household h at time t + i, Bt+i(h)are the household’s bond holdings, t+i is the nominal interest rate, nt+i(h) is the fraction of employed individuals within the household, G [et+i(h)]is the disu- tility of e¤ort - et+i(h) - of the typical working family member, V h

Mt+i(h) Pt+i

i is the utility arising from nominal money balances - Mt+i(h)- over the price level - Pt+i. Wt+i(h) and Tt+i(h) are the household’s nominal wage income and government transfers respectively. Finally, qt+i(h)are pro…ts that households receive from …rms.

3Feenstra (1986) showed the functional equivalence of money-in-the-utility-function models and liquidity-costs ones.

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In this framework, households, and not individuals, make all the decisions regarding consumption, bond holdings, real money balances and e¤ort⁴. In- dividuals are identical ex-ante, but not ex-post, given that some of them are employed - being randomly and costlessly matched with …rms independently from time - and some others are unemployed. The fraction of the unemployed is the same across all the families, and so their ex-post homogeneity is preserved.

Note that in our model no utility arises from leisure, therefore individual agents inelastically supply one unit of time for either work or unemployment related activities. Furthermore, after Akerlof (1982), workers, though dislik- ing e¤ort, will be ready to exert it as a gift to the …rm if they receive some other gift in exchange, such as a real compensation above some reference level.

Similarly to Danthine and Kurmann (2004), on the basis of the empirical evidence produced by Bewley (1998), we specify the e¤ort function, G [e_t+i(h)], as follows

G [e_t+i(h)] = (

e_t+i(h)

"

0+ ₁log^W_P^t+i^(h)

t+i + ₂log u_t+i(h)+

+ ₃log ^W_P^t+i

t+i + ₄log^W^t+i_P ¹^(h)

t+i

#)2

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in the personal norm case and as follows

G [et+i(h)] = (

et+i(h)

"

0+ ₁log^W_P^t+i^(h)

t+i + ₂log u_t+i(h)+

+ ₃log^W_P^t+i

t+i + ₄log ^W_P^t+i ¹

t+i

#)² (4)

4This modelling device is not only common to e¢ ciency wages models (Danthine and Kurmann, 2004, 2008, 2010), it is also used in neo-keynesian models with search frictions in the labour market (Blanchard and Galí, 2010 on the footsteps of Merz, 1995). Its underlying assumption is full risk sharing and its ultimate goal is to preserve a representative agent setup. Alexopoulos (2004) justi…es a similar framework assuming that households can observe individuals’behavior and that they can punish workers declining job o¤ers by withdrawing income insurance. It would also be possible to think that workers and not households decide how much e¤ort to elicit. However, since all workers within a household are symmetrical, it would not change our results.

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in the social norm case⁵. Wt+i is the aggregate nominal wage and ut+i(h) = 1 n_t+i(h)is the unemployment rate. Note that, with di¤erence to Danthine and Kurmann (2004), the nominal (either individual or aggregate) wage at time t + i 1 is assessed at the prices of time t + i. This assumption does not entail any money illusion. On the contrary, its underlying intuition is that households are aware of the damages that in‡ation can produce to their living standards and so they are ready to exchange more e¤ort for a pay policy that allows nominal wages to keep up with in‡ation. More brie‡y, a higher in‡ation rate reduces the reference wage.

Throughout the paper, similarly to Danthine and Kurmann (2004), we assume ₁; ₂ > 0 and ₃; ₄ < 0: In words a higher household’s real wage and a higher unemployment rate induce more e¤ort. On the other hand, a higher reference wage - be it due to either a higher aggregate wage or a higher real value of past compensation - depresses e¤ort.

Note that, under the hypothesis of an additively separable utility function, utility maximization implies that

G⁰[e_t+i(h)] = 0 (5)

and, therefore, that in the personal norm case e_t+i(h) = ₀+ ₁logW_t+i(h)

P_t+i + ₂log u_t+i(h)+ ₃log W_t+i

P_t+i + ₄logW_{t+i 1}(h) P_t+i

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5An alternative approach to the e¤ort function is the one pursued by Campbell (2006, 2008a and 2008b), which entails a more general functional speci…cation to be linearized at a later stage. However, calibration is less straightforward in this context and economic theorizing is usually followed by a number of numerical exercises where parameters and results display a somewhat large variation. For this reason we prefer to follow Danthine and Kurmann (2004).

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and in the social norm case e_t+i(h) = ₀+ ₁log W_t+i(h)

P_t+i + ₂log u_t+i(h) + ₃logW_t+i

P_t+i + ₄log W_{t+i 1} P_t+i

(7) Similarly to Danthine and Kurmann (2004), we assume that ct+i(h) and

Mt+i(h)

Pt+i enter (1) in logs

U ( ) = log c_t+i(h) n_t+i(h) G [e_t+i(h)] + b log M_t+i(h)

P_t+i (8)

Utility maximization implies

1

c_t+i(h) = E ^t+i

t+i+1

1

c_t+i+1(h) (9)

t+i t+i

1

= c_{t+i 1}(h)

c_t+i(h) 1 1

t+i

= 1 1

t+i 1

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where _t+i is the money growth rate and t+i is the in‡ation rate. The government rebates its seigniorage proceeds to households by means of lump- sum transfers, Tt(h):

Z1

0

T_t+i(h) P_t+i dh =

Z1

0

M_t+i(h) P_t+i dh

Z1

0

M_{t+i 1}(h)

P_t+i dh (11)

3 First variation: the social norm case

3.1 The long-run

Firms in the perfectly competitive product market hire individuals belonging to all the households to produce their output. Firms maximize their prof- its - Pt+iy_t+i

Z1

h=0

W_t+i(h)n_t+i(h)dh, where yt+i is output - subject to their

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production function - yt+i=hR1

0 e_t+i(h) ⁿⁿ¹n_t+i(h) ⁿⁿ¹ dhi ⁿ

n 1

, where n is the elasticity of substitution among di¤erent labour kinds - and to (7), by choosing nt+i(h) and Wt+i(h). Note that the production function displays decreasing marginal returns to each labour type and constant returns to scale.

The …rst order condition with respect to nt+i(h)equates the marginal cost of labour to its marginal product. All households are symmetrical, so we can drop the h index and write⁶

W_t+i

P_t+i = y_t+i

n_t+i (12)

whereas the …rst order condition with respect to Wt+i(h), instead, equates the marginal cost of rising the real wage to the bene…t that this induces by increasing e¤ort

W_t+i P_t+i

n_t+i y_t+i = ¹

e_t+i (13)

By substituting (12) into (13) ; one obtains the well known Solow condition

e_t+i = ₁ (14)

Therefore, …rms, maximizing their pro…ts, demand the same e¤ort to all households, across time and independently from the rate of in‡ation. Fur- thermore, (14) and the production function, under the condition of households’symmetry, imply

W_t+i

P_t+i = y_t+i

n_t+i = ₁ (15)

Substitute (14) and (15) into (7) and consider that trend in‡ation is equal to steady state money growth, , to obtain

log u = ⁰ ¹

2

+ ( ₁+ ₃+ ₄)

2

log ₁+ ⁴

2

log (16)

which, together with our standard assumptions on the sign of ₄ and ₂

6Equation (12) implies that qt(h) = 0.

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implies that the elasticity of the unemployment rate with respect to in‡ation is negative

d log u d log = ⁴

2

< 0 (17)

The intuition underlying this result is the following. An increase in in-

‡ation produces a decrease in the reference wage, by reducing the current real value of the past compensation. This would spur e¤ort, but the …rms’

optimal level of e¤ort does not depend on in‡ation. As a consequence …rms increase employment (and decrease unemployment) to keep the level of e¤ort constant. Following the results by Karanassou et al. (2005, 2008a, 2008b), one could calibrate ⁴

2 0:29.

Note that this mechanism does not imply that hyperin‡ation will produce large decreases in unemployment. In order to understand this point we focus on the semielasticity of the unemployment rate with respect to the money growth rate. In our context, the advantage of the semi-elasticity versus the elasticity is that it is a measure of the reactiveness of the unemployment rate to absolute, and not percentage, changes in the money growth rate, mirroring, under this respect, the results provided by, among others, Ascari (1998, 2004) and Graham and Snower (2004, 2008). The semielasticity of the unemployment rate with respect to money growth is

d log u

d = ⁴

2

1 < 0 (18)

which is still negative, given that 1, but lim _!1 ^{d log u}_d = 0.

3.2 The short-run

In order to analyze the short run dynamics of the present economic model, consider …rst that the only steady state condition we imposed to obtain (16) is the equality of money growth and in‡ation. Out of steady state one can write (16) as log ut+i = ⁰ ¹

2 +⁽ ¹⁺ ³⁺ ⁴⁾

2 log ₁+ ⁴

2 log _t+i. The other equations

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of the system are (9) ; (10) ; the aggregate resource constraint, yt = c_t; the production function and the condition nt = 1 u_t. The equilibrium for this model is a sequence fu^t+i; _t+i; n_t+i; y_t+i; _t+i; c_t+ig satisfying households’

utility maximization and …rms’pro…t maximization.

This system of equations, after log-linearization around the steady state, can be expressed as a second order di¤erence equation in in‡ation, which in its turn can be re-arranged to obtain the following system of …rst order di¤erence equations

E (^x_t+i+1) = 2 4 _ss+

uss

nss

1 + ^u_n^ss

ss

3 5 ^x_t+i

uss

nss ss

1 + ^u_n^ss

ss

^_t+i (19)

E (^_t+i+1) = ^x_t+i (20)

In the equations above, hats denote deviations from steady state, ⁴

2

and iss; u_ss and nss are the steady state values of the nominal interest rate, of the unemployment rate and of the employment rate respectively. In order to investigate the stability of (19)-(20) we need to calibrate not only ⁴

2 as above, but also iss; u_ss and nss. In order to do so we take as reference the averages of the post-second-world-war US time series and we set uss = 0:056, n_ss= 1 u_ss and iss= 1:02 (1 + ) :We compute the roots of (19)-(20) for various values of trend in‡ation and the results are showed in Figure 1. As it is possible to see the system is always saddle-path stable, give that one root is outside the unit circle and the other one within it.

It is possible to wonder what are the e¤ects of trend in‡ation on the stable arm of the system. The answer to this question is showed in Figure 2 where, following Shone (2001), di¤erent trajectories along the stable arm are projected on the f ^t; _t+1g plane for trend in‡ation rates equal to 2%, 20% and 80%. The higher is trend in‡ation and the ‡atter is the stable arm.

In other words, the higher is trend in‡ation and the sharper should in‡ation reductions be in order to achieve stability.

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4 Second variation: the personal norm case

In the personal norm case, …rms recognize that wage setting has intertemporal consequences. A wage increase will induce more e¤ort in the …rst period by raising the household’s real wage, but it will decrease e¤ort in the second period by raising the household’s reference wage. The …rms’ pro…t maximization problem will therefore be

max

fn^t+i(h);Wt+i(h)g

X1 j=0

t;t+i

2

4P_t+iy_t+i Z1

h=0

W_t+i(h)n_t+i(h)dh 3 5

s:t: y_t+i = Z 1

0

e_t+i(h) ⁿⁿ¹n_t+i(h) ⁿⁿ¹ dh

n n 1

(21) e(h) = ₀+ ₁logW_t+i(h)

P_t+i + ₂log ut+i(h) + ₃log W_t+i

P_t+i + ₄logW_{t+i 1}(h) P_t+i where t;t+i is the …rm discount factor.

In the present setting (13) turns out to be

t;t+in_t+i(h) = _t;t+i y_t+i e_t+i(h)

1 Wt+i(h)

Pt+i

! +E

2

4 _t;t+i+1 y_t+i+1 e_t+i+1(h)

0

@ _W ⁴

t+i(h) Pt+i t+i+1

1 A 3 5 (22) In words, …rms equate the discounted marginal cost of increasing the real wage to the sum of its discounted marginal revenues, which are composed by a positive e¤ort e¤ect in period t + i and a negative e¤ort e¤ect in period t + i + 1.

Consider that households and …rms have access to a complete set of fric- tionless security markets, which, after Lucas (1978) and Collard and de la Croix (2000), implies that, at equilibrium, t;t+i will be proportional to the discounted marginal value of wealth, which, assuming a logarithmic separable utility function in consumption and knowing that ct+i = yt+i; will be

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equal to ^t+i=y_t+i:

Substituting (12) which holds also for the present model - into the previous equation and re-arranging one has

1 = ¹

e_t+i(h)+

e_t+i+1(h) ⁴ ^t+i+1 (23)

In steady state this implies a modi…ed Solow condition, which, after drop- ping the h index due to symmetry, is

e = ₁+ ₄ (24)

Firms still demand the same e¤ort level to all households across time, but not independently from money growth, given that they now take into account its discounted future e¤ect on e¤ort. As a consequence trend in‡ation appears to have a negative impact on …rms’ desired level of e¤ort. This happens because there are diminishing returns to the e¤ort connected to h th kind of labour input. Under such circumstances trend in‡ation, equal to trend money growth, would induce households to elicit more e¤ort in time t + i + 1 by reducing the reference wage. However, under diminishing returns, this is less and less bene…cial to …rms and, as a consequence, the marginal revenue to wage increases would fall below their marginal cost. Firms, therefore, anticipate households’behavior by demanding less e¤ort to each household the greater is money growth. Due to symmetry, this produces a negative link between trend in‡ation and aggregate e¤ort⁷.

This implies that the e¤ect of money growth on unemployment does not vanish at high in‡ation rates. Along the lines followed in the previous section

7A graphical account of this intuition is set out in Figure A1 in the Appendix, where (23) is depicted. The left hand side of (23) is the marginal cost of rising wages per unit of discounted labour. The right hand side, instead, is the marginal bene…t, which is a decreasing function of the e¤ort level because there are diminishing returns to the e¤ort elicited by household h. Money growth reduces the marginal revenue to rising wages and therefore shifts inward the marginal revenue schedule, producing a fall in the desired level of e¤ort, that balances the marginal revenue and cost to a wage increase.

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it is easy to show that in the present model (18) turns out to be d log u

d = ⁴

2

( ₁+ ₃ + ₄)

2

4

( ₁+ ₄ ) + ⁴

2

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As a consequence lim _!1 ^{d log u}_d 6= 0, because …rms hire more workers in the attempt to reduce e¤ort as money growth rises. This is unrealistic and we will not develop the present model any further.

5 Third variation: the social norm case with wage staggering

5.1 The long-run

In the present section we combine e¢ ciency wages with Taylor wage staggering. In order to do so we assume households to belong to di¤erent cohorts, whose labour services are not perfect substitutes. This assumption is necessary because if di¤erent labour kinds were perfect substitutes, labour demand for cohorts whose wage is reset would go to zero. The wage is not set by households, as usual in wage staggering model, but by …rms, as customary in fair wages models.

Note that, due to the existence of wage staggering, households belonging to di¤erent cohorts have di¤erent income levels. However, as customary, we assume they have access to complete asset markets, which allows them to consume all the same amount of the …nal good as implied by the …rst order condition with respect to consumption in problem (1) (2).

Following Graham and Snower (2004), one can write the …rms’ pro…t

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maximization problem as follows

max fⁿ^t+i^(h);W^{t+N j}^(h)g

X1 j=0

(j+1)NX 1 i=jN

t;t+i

2 4y_t+i

Z1

h=0

Wt+N j(h)

P_t+i n_t+i(h)dh 3 5

s:t: y_t+i = Z 1

0

n n 1

(26) e(h) = ₀+ ₁logW_{t+N j}(h)

P_t+i + ₂log u_t+i(h) + ₃logW_t+i

P_t+i + ₄log W_{t+i 1} P_t+i where N is the contract length. The …rst order conditions with respect to n_t+j(h)and Wt+N j(h)and the recursiveness of the problem above imply

W_t(h) P_t+i = y

1 n

t+ie_t+i(h) ⁿⁿ¹n_t+i(h) ¹ⁿ (27)

NX1 i=0

t;t+in_t+i(h) =

NX1 i=0

t;t+i

Z 1 0

n n 1 1

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e_t+i(h) ⁿⁿ¹ ¹n_t+i(h) ⁿⁿ¹ ¹ W_t(h)=P_t+i Substituting (27) into (28) one obtains

N 1

X

i=0 t;t+i

n_t+i(h)

P_t+i 1 ¹

e_t+i(h) = 0 (29)

which, given that t;t+i; n_t+i(h); P_t+i > 0, leads to the Solow condition

e_t+i(h) = ₁ (30)

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Substituting (30) into yt+i =hR1

0 e_t+i(h) ⁿⁿ¹n_t+i(h) ⁿⁿ¹ dhi ⁿ

n 1

one has

1 = 1

1

(Z 1 0

W_t(h) P_t+i

1 n

dh )₁ ¹

n

(31)

and in steady state W

P = ₁ 1

N

1 ^{N (} ⁿ ¹⁾

1 ⁿ ¹

1 n 1

(32)

where W is the reset wage.

Further, substitute the Solow condition into (7) and aggregate across households keeping in mind that _P^P^t+i

t+i 1 = to obtain

1 = ₀+ ₁

NX1 j=0

log ^W_P ^j

N + ₂log u + ( ₃+ ₄) logW

P ⁴log (33) and

log u_{W S} = ¹ ⁰

2

1 2

log (

1

1 N

1 ^{N (} ⁿ ¹⁾

1 ⁿ ¹

1 n 1)

+ (34)

+ ¹

2

(N 1)

2 log + ⁴

2

log ( ₃+ ₄)

2

log W P where the subscript W S stays for wage-staggering.

Subtracting (34) from (16) and taking the …rst order derivative with respect to , one can compute the semielasticity of the percentage deviation of the unemployment rate under wage staggering from its level with ‡exible

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wages

@ (log u_{W S} log u)

@ = ¹

2

N [1 ^{N (} ⁿ ¹⁾]

N ( n 1) 1+ + ¹

2

1 [1 ⁽ ⁿ ¹⁾]

( n 1) 1 (35)

1 2

(N 1) 2

1

If ^@(^{log u}_{W S} ^{log u}f f)

@ is negative, it will mean that unemployment will be more responsive to absolute changes in money growth under wage staggering than under ‡exible wages. In order to explore this issue, it is necessary to check that the following condition holds⁸

( ) = N

[1 ^{N (} ⁿ ¹⁾]

N ( n 1)+ 1

[1 ⁽ ⁿ ¹⁾]

( n 1) (N 1)

2 > 0 (36) We do so for di¤erent values of N and n in Figures 3 and 4 respectively.

In both the cases (36) is veri…ed.

The intuition for this result is that wage staggering has two e¤ects on e¤ort. On the one hand, wage dispersion increases with in‡ation, leading to a higher ratio between the wage of the resetting cohort and the aggregate wage index. On the other hand, a higher in‡ation rate means that, over the contract period, the real wage of not-resetting cohorts will decline faster. The former e¤ect has a positive impact on e¤ort, while the latter a negative one.

However, the former prevails on the latter one. As a matter of consequence

…rms have to increase employment and decrease unemployment to a greater extent than under ‡exible wages in order to keep e¤ort at their constant desired level. Increasing N and n boosts wage dispersion, decreasing the slope of the long-run Phillips curve.

8Recall that ¹

2 < 0.

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5.2 The short run

In order to analyze the short run dynamics of the present economic model, we set N = 2. The equation for the log of the unemployment rate can be obtained integrating the e¤ort function over h and keeping in mind equation (31) :

log u_t+i= ⁰ ¹

2

+ ¹

2

Z 1=2 0

logW_t+i(h)

P_t+i dh+ ¹

2

Z 1 1=2

log W_{t+i 1}(h)

P_{t+i 1} _t+idh ⁴

2

log _t+i (37)

The other equations of the system are (9) ; (10) ; (31) ; the aggregate resource constraint - yt = c_t -, the de…nition of unemployment rate R1=2

0 n_t(h)dh + R1

1=2n_t(h)dh = 1 u_t, and the demands for the labour services of the households belonging to the two cohorts:

W_t+i(h) Pt+i

= y_t+i nt+i(h)

1 n

for h 2 0;1

2 (38)

W_{t+i 1}(h) Pt+i 1 t+i

= y_t+i nt+i(h)

1 n

for h 2 1

2; 1 (39)

Finally, the autoregressive process for money growth is

t= ¹ _{t 1}exp( _t) (40)

The equilibrium for this model is a sequence n_W

t+i(h)

Pt+i ; _t+i; u_t+i; _t+i; nt+i(0); nt+i(1); yt+i; t+i; ct+ig satisfying households’utility maximization and

…rms’ pro…t maximization. We log-linearized the system around a steady state with uss = 0:056 on the basis of the US post-WWII experience. We calibrated the system parameters as customary in the New-Keynesian literature (see for instance Ascari, 2004): = 1:04 ¹²; = 1:02¹²; _n = 5,

4

2 = 0:29, = 0:57¹². In order to attach a value to ¹

2 we note that it can be considered as the inverse of the elasticity of households’wages with respect

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to the unemployment rate and so we set it to 0:07 ¹after Nijkamp and Poot (2005).

Figure 5, as similar …gures below, plots the percentage deviations from steady state of the in‡ation rate against those of the unemployment rate.

In other words, we plot the impulse response function of the in‡ation rate against that of the unemployment rate in order to show the unemployment- in‡ation trade-o¤ in a more direct way. As it is possible to see, wage staggering implies a ‡atter Phillips curve than ‡exible wages not only in the long-run but in the short run too. Note that increasing n from 5 to 15 would not change our results markedly⁹. Instead, increasing N from 2 to 4 has a considerable impact on the dynamics of in‡ation and unemployment. As showed in Figure 6, their reactiveness increases, however, unemployment …rst declines and then increases before going back to its steady state value. A shortcoming of this model is that, with di¤erence to the other models presented in this work, a monetary expansion can cause a contraction in output due to the ine¢ ciencies arising from …rms shifting labour demand from one cohort to the other, given that di¤erent labour kinds are imperfect substitutes. For N=4 and n = 5 a one percentage shock in money growth produces a 0.18 percent decline in output. This is implausible and for this reason the model presented in this section is not our preferred one.

6 Fourth variation: the social norm case with varying capital

Once considering varying capital within the model, we assume the existence of capital adjustment costs after Bernanke et al. (1999) and Gertler (2002).

The households’budget constraint changes to

9Further results are available from the author upon request.

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c_t+i(h) = W_t+i(h)

P_t+i n_t+i(h) + T_t+i(h) P_t+i

M_t+i(h)

P_t+i + M_{t+i 1}(h) P_t+i

B_t+i(h)

P_t+i + (41)

+B_{t+i 1}(h)

P_t+i ^{t+i 1}+R_t+i

P_t+iK_t+i(h) Q_t+i

P_t+i [K_t+i(h) (1 ) K_{t+i 1}(h)] + q_t+i(h) where Kt+i(h) is the capital held by household h, is the capital deprecia-

tion rate, Rt+i is the capital rental rate and Qt+i is the nominal Tobin’s q.

Furthermore, households maximize utility with respect to capital too and interacting the …rst order conditions for capital and consumption leads, under households’symmetry, to the following equation

E(ct+i+1

Q_t+i

P_t+i) = R_t+i

P_t+iE(ct+i+1) + ct+i (1 )Q_t+i+1

P_t+i+1 (42)

As in the New-Keynesian tradition, we assume the existence of an intermediate labour market, where labour intermediaries hire households’horizon- tally di¤erentiated labour inputs to produce homogeneous labour to be sold to …rms operating on the …nal product market. In the intermediate labour market we assume productivity to depend on e¤ort. The pro…t maximization problem of labour intermediaries is

max

fn^t+i(h);Wt+i(h)g

W_t+in_t+i Z 1

0

W_t+i(h)n_t+i(h)dh (43)

s:t: n_t+i= Z 1

0

e_t+j(h) ⁿⁿ¹n_t+j(h) ⁿⁿ¹ dh

n n 1

The solution of this problem and households’symmetry imply W_t+i(h)

Wt+i

= n_t+i

nt+i(h) = ₁ = e_t+i = 1 (44) Firms in the …nal product market maximize pro…ts hiring labour and

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capital and adopting a Cobb-Douglas production function. The solution of their problem leads to two customary demand functions for labour and capital

(1 )y_t+i

Wt+i

Pt+i

= n_t+i (45)

yt+i Rt+i

Pt+i

= K_t+i (46)

Substituting these two equations into the production function one has W_t+i

P_t+i =

Rt+i

Pt+i

! ₁

(1 ) (47)

Finally, capital producer j has the following production function

Y_t+i^k (j) = I_t+i(j)

K_{t+i 1}(j) K_t+i(j) (48)

where Y_t+i^k (j) is new capital, It+i(j) is raw output used as material input at time t + i and ⁰( ) > 0, ⁰⁰( ) < 0, (0) = 0 and _KÎ = _KÎ, with _KÎ being the steady state investment-capital ratio. Kt+i(j) is capital rented after it has been used to produce …nal output within the period. The pro…ts of the j-th capital producer can be written as ^Q_P^t+i

t+i

h It+i(j) Kt+i 1(j)

i

K_t+i(j) I_t+i(j) Z_t+i^k K_t+i(j) where Z_t+i^k is the rental price of capital used for producing new capital. The …rst order condition for It+i(j) is, under a symmetry condition:

Q_t+i P_t+i

0 I_t+i

K_{t+i 1} 1 = 0 (49)

where It+i=R1

0 I_t+i(j) dj and Kt+i 1 =R1

0 K_{t+i 1}(j) dj. One can show that the …rst order condition with respect to Kt+i(j) ; _K^I = _K^I and (49) imply that Z_t+i^k is approximately zero near the steady state and so it can be ignored.

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The system of equations is therefore composed by (9) ; (10) ; the aggregate resource constraint yt+i = c_t+i+ I_t+i, the law of motion of capital Kt+i =

K It+i

Kt+i 1 K_t+i (1 ) K_{t+i 1}, the de…nition of the unemployment rate n_t = 1 u_t, (40), (42), (45) ; (46), (47), (49) and (7), which imposing (44) and after rearranging becomes

log u_t+i= ⁰ ¹

2

4 2

log _t+i+( ₁+ ₃)

2

logW_t+i P_t+i + ⁴

2

logW_{t+i 1} P_{t+i 1} (50) The equilibrium of this system is a sequencen

Rt+i

Pt+i;^W_P^t+i

t+i; y_t+i; n_t+i; K_t+i; c_t+i; u_t+i; _t+i; _t+i; _t+i; I_t+i;^Q_P^t+i

t+i

o

satisfying utility and pro…t maximization prob- lems.

Regarding the long-run we note that in steady state the real Tobin’s q is equal to one and therefore that ^R_P and ^W_P are pinned down by (42) and (47) independently from money growth. On the basis of (50) and of the steady state equality of in‡ation and money growth, this entails that (17) and (18) also hold for the present model.

Regarding the short-run, we do not change the calibration of the parameters that already appeared in the previous sections of the present work, with the only exception that, given that we have ‡exible wages here, we do not rise them to the power of ¹₂. Following the same reasoning above regarding the elasticity of the wage to the unemployment rate we set ⁽ ¹⁺ ³⁾

2 = 0:07 ¹. Furthermore, as customary, = 0:33, = 1 0:92 and, after Bernanke et al. (1999), = ⁰⁰[_K^I]_K^I

0[_K^I] = 0:5: We log-linearize the system around the steady state. The short-run Phillips curve with …xed and varying capital are plotted in Figure 7. The result that higher in‡ation goes hand in hand with a lower unemployment rate, whose intuition was discussed commenting equation (17), is con…rmed also for the present model. As it is possible to see, varying capital implies a ‡atter short run Phillips curve than under …xed capital, given that the boom following a monetary expansion is reinforced by

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an increase of investments, which rise upon impact by 0.08%¹⁰. Figure 7 also shows that increasing trend in‡ation decreases the responsiveness of both the in‡ation and unemployment rates to a 1% monetary shock. This is consistent with our results above that increasing trend in‡ation ‡attens the stable arm of the economic system without capital and it can be explained by keeping in mind two facts. First, households smooth consumption and, second, an increase in trend in‡ation decreases the elasticity of the money demand function to the nominal interest rate¹¹. If households smooth consumption, they will tend to smooth also real money holdings - see equation (10). This, in presence of a smaller reactiveness of money demand to the nominal interest rate, can happen only thanks to a larger reaction in the latter one (Figure 8). In other words, households achieve a stable path for consumption and real money holdings in face of a monetary shock with higher trend in‡ation by letting the interest rate to react more, which stabilizes the whole economy and implies a smaller change in in‡ation too. A smaller change in the in‡ation rate translates into a smaller change in the unemployment rate via the Phillips curve (50).

10Changing would only have negligible e¤ects on the Phillips curve. Further results are available from the author on request. It is worth noting that our model does not produce a persistent reaction of either the unemployment or the in‡ation rate after a monetary shock. This accords well with the empirical evidence produced by the in‡ation persistence network, whose main result is that, once allowing for structural breaks in the mean of the in‡ation time series, in‡ation has low persistence (Altissimo et al., 2007).

Empirical evidence of a fast adjustment of unemployment after a monetary shock was produced by Karanassou et al. (2007, p. 346) where the unemployment rate takes just two periods to hit its new long-run level after a permanent monetary shock. However, this low persistence is not a property of e¢ ciency wages themselves. Danthine and Kurmann (2004, 2010) showed that, once e¢ ciency wages are coupled with price rigidities, it is possible to produce persistent impulse response functions.

11Loglinearizing (10) ; one can show that this elasticity is _i ¹

ss 1 where iss is equal to trend in‡ation over the discount factor.

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7 Fifth and sixth variations: reciprocity in labor relations and the Phillips curve

The present section adopts an approach à la Danthine and Kurmann (2008, 2010), that can be nested into our model by specifying G [et+i(h)] = ¹₂[e_t+i(h)]²

< [e^t+i(h); :] ; where < [e^t+i(h); :] is the product of the gifts of the representative worker, d [et+i(h); :] ; and the …rm g [Wt+i(h); :]. In words, when per- ceiving a generous wage o¤er by the …rm - g [Wt+i(h); :] > 0 - the utility of a worker increases by eliciting more e¤ort - d [et+i(h); :] > 0. Note that d [e_t+i(h); :] = [e_t+i(h)] with 0 < < 1 and

g [Wt+i(h); :] = log W_t+i(h)

P_t+i f₁log y_t+i

n_t+i(h) f₂logW_t+i

P_t+int+i (51) f₃log (1 s) W_{t+i 1}(h)

P_t+i + s W_{t+i 1} P_t+i In the above equation, log^W_P^t+i^(h)

t+i accounts for the the consumption utility at- tached by the representative worker to the …rm’s actual wage o¤er. log_n^y^t+i

t+i(h)

proxies for …rms’ability to pay, by describing the utility obtained if the …rm distributed its whole revenue to workers. In case a worker quits and …nds a job elsewhere, s/he will enjoy the expected utility log^W_P^t+i

t+in_t+i. Finally, logn

(1 s)h_W

t+i 1(h) Pt+i

i

+ s ^W_P^t+i ¹

t+i

o

represents the e¤ect of the current real value of past compensation on the reference wage. This formulation encom- passes both the social norm case (with s = 1) and the personal norm one (with s = 0). Finally, f1; f₂ and f3 are non-negative parameters.

The condition G⁰[e_t+i(h)] = 0 here implies the following e¤ort function

e_t= ²¹ 0

@ log^W_P^t+i^(h)

t+i f₁log _n^y^t+i

t+i(h) f₂log^W_P^t+i

t+in_t+i f₃logn

(1 s)h

Wt+i 1(h) Pt+i

i

+ s ^W_P^t+i ¹

t+i

o 1 A

1 2

(52)

We perform the …rm pro…t maximization problem as in our second vari-

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ation above to obtain the …rst order conditions with respect to Wt+i(h) and n_t+i(h): Under household symmetry they are

W_t+i

P_t+i = y_t+i

n_t+i 1 + e_t+i² 1

2 f₁ (53)

t;t+in_t+i = 8>

><

>>

:

t;t+iyt+i

et+i

1

2 e_t+i¹_Wt+i

Pt+i

t;t+i+1yt+i+1

et+i+1

1

2 e_t+i+1¹ _Wt+i^f³

Pt+i t+i+1

(_{1 s}¹ ) 9>

>=

>>

;

(54)

After some manipulation and interacting (53) and (54) ; one obtains a similar steady state equation to (24) :

e = 1

2 1 f₃

1 s f₁

1 2

(55) Along the lines followed in the previous variations it is possible to show that lim _!1^{@ log n}_@ 6= 0: In other words, one obtains the same result as in our second variation at the price of a heavier parametrization.

We now focus on the social norm case, namely we impose s = 1. Under this assumption (54) and (55) change into

nt+i = 2

y_t+ie_t+i²

Wt+i

Pt+i

(56)

e_t+i = (1 f₁) 1 2

1 2

(57)

On the footsteps of our …rst variation, it is possible to show that the employment rate and the in‡ation rate are linked by the following equation

log n_t+i=constant+f3

f₂ log _t+i (58)

This has similar implications for the short-run and long-run Phillips curve to